An important tool we leverage to establish our results in the universal learning setting is the theory of Gale-Stewart (GS) games [1]}. Every such game consists of two players, a learner \(\texttt {P}_L\)  and an adversary \(\texttt {P}_A\) , and is played over an infinite sequence of discrete steps. In each step, the adversary presents some point \(x_t \in \mathcal {X}_t\)  to the learner and the learner
picks a response \(y_t\in \mathcal {Y}_t\) .
If some predefined condition gets violated at some step \(t\) , the game terminates and the learner wins. On the other hand, if the condition does not get violated during the infinite sequence of these time-steps, the adversary wins. The main property which characterizes the GS game is that the winning strategy of the learner is finitely-decidable, i.e., she knows that she has won the game after playing a finite number of rounds.
[1]}, [3]}, [4]} proved that either \(\texttt {P}_L\)  or \(\texttt {P}_A\) 
has a winning strategy, i.e., playing that strategy makes them win regardless of the opponents actions.
Similar to [5]}, the main reason we use GS games in this work is to obtain functions that are eventually correct. We deal with GS games that are finite but there is not a bound on
the number of steps they need to terminate. It turns out that an ordinal is the right notion
to use for measuring the
remaining steps needed for the game to terminate.
The key motivation behind the use of ordinal numbers is to capture the intermediate state between an infinite tree
and a uniformly bounded tree.
